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I studied this at the beginning of my graduate degree but have to review it for my graduate exam. If it's not clear I'm talking about the $\beta = \frac{it}{\hbar} $ turning the integral of your propagator into a "partition function" and the ensuing analysis of eigenvalues as poles in the complex plane. I remember this as being a particularly beautiful and powerful approach but one that was difficult to follow the details and keep an intuitive picture of what's going on.

I'd like to find books (or articles or websites) that have a good qualitative and quantitative explanation of what's going on without getting too bogged down with the mathematical details. Shorter is better but not at the expense of clarity.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

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Hi mmdanziger, and welcome to Physics Stack Exchange! While this is a good thing to ask about, it makes a better question if you actually ask the question you have about the physics itself, rather than asking for a book that describes it. People can still answer by referring you to a book if that is appropriate. – David Z Feb 14 '12 at 19:54

For a path integral formulation of this technique, I recommend Hagen Kleinert's massive book on path integrals, if you have access to a copy, or Jean Zinn-Justin's book on path integrals, which is shorter and also very good.

Of course the $\beta = i t / \hbar$ technique can be used in other formulations of quantum mechanics, e.g. the standard canonical quantization. The technique you're looking for though goes under the name of euclidean quantum mechanics. Even just a quick search online turned up many potentially useful sources. Take a look and find one really like.

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Since this has become a community wiki... I think that the following two references (authors mentioned below) are among the standard recommendations for thermal field theory:

  1. Kapusta & Gale
  2. Le Bellac

If I recall correctly, the former uses the imaginary-time formalism while the latter also has a treatment of the real-time formalism.

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